| L(s) = 1 | + (1.57 − 4.95i)3-s − 8·4-s − 14.3i·5-s + 5.45·7-s + (−22.0 − 15.6i)9-s + (−12.6 + 39.6i)12-s + (−71.0 − 22.6i)15-s + 64·16-s + 61.4i·17-s − 159.·19-s + 114. i·20-s + (8.59 − 27.0i)21-s − 80.8·25-s + (−111. + 84.4i)27-s − 43.6·28-s + 3.27i·29-s + ⋯ |
| L(s) = 1 | + (0.303 − 0.952i)3-s − 4-s − 1.28i·5-s + 0.294·7-s + (−0.816 − 0.578i)9-s + (−0.303 + 0.952i)12-s + (−1.22 − 0.389i)15-s + 16-s + 0.876i·17-s − 1.92·19-s + 1.28i·20-s + (0.0893 − 0.280i)21-s − 0.646·25-s + (−0.798 + 0.602i)27-s − 0.294·28-s + 0.0209i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.117867 + 0.758907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117867 + 0.758907i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.57 + 4.95i)T \) |
| 59 | \( 1 + 453. iT \) |
| good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 + 14.3iT - 125T^{2} \) |
| 7 | \( 1 - 5.45T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 61.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 3.27iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 450. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.3iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.18e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.39e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.23481228606290600691422484774, −10.69102150833420392225325246239, −9.221034150226689550101459524693, −8.567013926483412586395556568972, −7.941712549700444723198189741399, −6.29291062911492051506644527401, −5.08288369691708611030722406455, −3.93232725477115576020955822793, −1.76306720346853333465005933389, −0.34200826286994347660510959214,
2.70309303190026504760401059898, 3.94397026756928190544901243350, 4.96808870443704163107983524371, 6.38726489432904388152097064027, 7.88568581792547979547884420356, 8.850156423830596864415932079742, 9.868583732353880663384382388751, 10.61301895567652623012931867625, 11.47094680310679674245537816486, 12.99709166017941076698722672525
